Quantum transitions of d-wave superconductors in a magnetic field Eugene Demler (Harvard) Kwon Park Anatoli Polkovnikov Subir Sachdev Matthias Vojta (Augsburg) Ying Zhang Science 86, 479 (1999). Transparencies online at http://pantheon.yale.edu/~subir
Further neighbor magnetic couplings La CuO4 T=0!" S 0 Experiments Magnetic order Universal properties of magnetic quantum phase transition do not change along this line!" S = 0 Concentration of mobile carriers δ in e.g. La Sr CuO δ δ 4 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 411 (199). A.V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, 11919 (1994)
Outline I. Magnetic ordering transitions in two-dimensional quantum antiferromagnets: NMR measurements of Zn/Li impurities and neutron scattering measurements of phonon spectra. II. III. Effect of magnetic field on antiferromagnetic order in superconductor. Comparison of predictions with neutron scattering experiments. Conclusions
I. Magnetic quantum transition in the insulator (δ=0) φ α α =1,,3 Neel order parameter Action: S b 1 = d xdτ φ + c φ + V φ Missing: Spin Berry Phases (( ) ( ) ) ( ) x α τ α α S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 344 (1989). A isa e Berry phases induce bond charge order in quantum disordered phase with φ α = 0 ; e.g. spin-peierls or plaquette order (need not be quasi-1d) Dual order parameter N. Read and S. Sachdev, Phys. Rev. Lett. 6, 1694 (1989).
Square lattice with first(j 1 ) and second (J ) neighbor exchange interactions H = J S " S " i< j ij i j 1 = ( ) N. Read and S. Sachdev, Phys. Rev. Lett. 6, 1694 (1989). O. P. Sushkov, J. Oitmaa, and Z. Weihong, Phys. Rev. B 63, 10440 (001). M.S.L. du Croo de Jongh, J.M.J. van Leeuwen, W. van Saarloos, Phys. Rev. B 6, 14844 (000). Neel state Spin-Peierls (or plaquette) state Bond-centered charge order See however L. Capriotti, F. Becca, A. Parola, S. Sorella, cond-mat/010704. Onset of bond-charge order: Z 4 order parameter Co-existence Vanishing of spin gap and Neel order: φ 4 field theory J / J 1
Properties of paramagnet with bond-charge-order Stable S=1 spin exciton quanta of 3-component φα ε k = + ck + ck x x y y Spin gap S=1/ spinons are confined by a linear potential. S=0 holes can similarly be confined in pairs. E. Fradkin and S. Kivelson, Mod. Phys. Lett B 4, 5 (1990). S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 19 (1991).
Effect of static non-magnetic impurities (Zn or Li) Zn Zn Zn Zn Spinon confinement implies that free S=1/ moments form near each impurity χ impurity ( T 0) = SS ( + 1) kt 3 B A.M Finkelstein, V.E. Kataev, E.F. Kukovitskii, G.B. Teitel baum, Physica C 168, 370 (1990). J. Bobroff, H. Alloul, W.A. MacFarlane, P. Mendels, N. Blanchard, G. Collin, and J.-F. Marucco, Phys. Rev. Lett. 86, 4116 (001) D. L. Sisson, S. G. Doettinger, A. Kapitulnik, R. Liang, D. A. Bonn, and W. N. Hardy, Phys. Rev. B 61, 3604 (000).
Further neighbor magnetic couplings La CuO4 Magnetic order!" S 0 Experiments Concentration of mobile carriers δ in e.g. T=0 La!" S = 0 Sr CuO δ δ 4 Summary: Confined, paramagnetic Mott insulator has 1. Stable S=1 spin exciton φ α. (resonance mode). Broken translational symmetry:- bond charge order. (phonon spectra) 3. Pairing of holes. 4. S=1/ moments near non-magnetic impurities. (NMR experiments) Properties of paramagnetic insulator survive for a finite range of δ
Neutron scattering measurements of phonon spectra Discontinuity in the dispersion of a LO phonon of the O ions at wavevector k = π/ : evidence for bond-charge order with period a k = 0 k = π La 1.85 Sr 0.15 CuO 4 Oxygen Copper YBa Cu 3 CuO 6.95 R. J. McQueeney, T. Egami, J.-H. Chung, Y. Petrov, M. Yethiraj, M. Arai, Y. Inamura, Y. Endoh, C. Frost and F. Dogan, cond-mat/0105593. R. J. McQueeney, Y. Petrov, T. Egami, M. Yethiraj, G. Shirane, and Y. Endoh, Phys. Rev. Lett. 8, 68 (1999). L. Pintschovius and M. Braden, Phys. Rev. B 60, R15039 (1999).
Computation of phonon damping by non-linear coupling to fluctuating spin-peierls order with period a K. Park and S. Sachdev, cond-mat/0104519
Further neighbor magnetic couplings T=0!" S = 0 La CuO4!" S 0 Experiments Magnetic order Concentration of mobile carriers δ in e.g. La Sr CuO δ δ 4 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 411 (199). A.V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, 11919 (1994) Universal properties of magnetic quantum phase transition do not change along this line
III. Effect of magnetic field on SDW order in SC phase T=0, H=0 Superconductor (SC) + Spin density wave (SDW) Superconductor (SC) r c r (some parameter in the Hamiltonian, possibly carrier concentration) Many experimental indications that the cuprate superconductors are not too far from such a quantum phase transition: G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science 78, 143 (1997). Y. S. Lee, R. J. Birgeneau, M. A. Kastner et al., Phys. Rev. B 60, 3643 (1999). S. Katano, M. Sato, K. Yamada, T. Suzuki, and T. Fukase Phys. Rev. B 6, 14677 (000). B. Lake, G. Aeppli et al., Science 91, 1759 (001). Y. Sidis, C. Ulrich, P. Bourges, et al., cond-mat/0101095. H. Mook, P. Dai, F. Dogan, cond-mat/010047. J.E. Sonier et al., preprint.
Main results (extreme Type II superconductivity) T=0 Elastic scattering intensity I( H) H 3H 1 a ln I(0) H H c = + c H ( r rc ) ~ ln 1/ ( ( r rc )) S = 1 exciton energy ε ( H) ε(0) H 3H 1 b ln H c = c H All functional forms are exact. Similar results apply to other competing orders e.g. SC + staggered flux E. Demler, S. Sachdev, and Y. Zhang, Phys. Rev. Lett. in press, cond-mat/010319.
Structure of quantum theory Charge-order is not critical: can neglect Berry phases. Generically, momentum conservation prohibits decay of S=1 exciton φ α into S=1/ fermionic excitations at low energies. Virtual pairs of fermions only renormalize parameters in the effective action for φ α. Zeeman coupling only leads to corrections at order H Simple Landau theory couplings between φ α and superconducting order ψ are allowed (S.-C. Zhang, Science 75, 1089 (1997)), e.g.: V φ V φ + λφ ψ S b ( ) ( ) α α α 1 = d xdτ φ + c φ + V φ (( ) ( ) ) ( ) x α τ α α
Theory should account for quantum spin fluctuations All effects are ~ H except those associated with H induced superflow. Can treat SC order in a static Ginzburg-Landau theory Action F / T + S + S GL b c 4 ψ FGL = d x ψ + + x ia ( ) ψ v Sc = d xdτ φα ψ S b 1/ T 1 g 0 ( ( ) ( ) ) ( ) x α τ α α α = d x dτ φ + c φ + rφ + φ 4!
Self-consistent Hartree theory of quantum spin fluctuations (large N limit) ( xx,, ) = ( x, ) ( x, ) χ ω δ φ ω φ ω n αβ α n β n ( ω ) n c x + V ( x) χ( x, x, ωn) = δ( x x ) V ( x) ( ) ( / 6 ) (,, ) = r+ v ψ x + NgT χ x x ω ( ) ( ) " 1 + ψ x x ia ψ( x) + ( NvT / ) χ( x, x, ωn) ψ( x) = 0 V ( x) local classical energy of spin fluctuations; can become negative in vortex cores for v > 0. However, spin gap remains finite because of quantum fluctuations ω ω n n n
Energy Lowest energy spin-exciton eigenmode V ( x) # Spin gap 0 x Vortex cores As 0, #, because of self interaction, g, of spin excitations. A.J. Bray and M.A. Moore, J. Phys. C 15, L765 (198). J.A. Hertz, A. Fleishman, and P.W. Anderson, Phys. Rev. Lett. 43, 94 (1979).
Dominant effect: uniform softening of spin excitations by superflow kinetic energy r v s 1 r Spatially averaged superflow kinetic energy H 3H v ln c s Hc H See D. P. Arovas et al., Phys. Rev. Lett. 79, 871 (1997) for a different viewpoint.
( x) Influence of ψ on extended spin eigenmodes: ψ ψ 1 x ( x) = 1 outside each vortex core because ( x) In SC phase, spin gap obeys: of superflow kinetic energy 3.0H c = 1 ln Hc H 3.0H c ( H ) = ( 0) ln 3v Hc H Ng 1 g In SC+SDW phase, intensity of elastic scattering obey s: H 4π c v 6v H 3.0H = + 3v Hc H g 1 g c ( ) I( 0) ln I H H
Intensity (counts/min) 35 30 5 Elastic neutron scattering off La CuO 4+ y B. Khaykovich, Y. S. Lee, S. Wakimoto, K. J. Thomas, M. A. Kastner, and R.J. Birgeneau, preprint. H = 7.5 T k H = 0 T h H c I(H)/I(0) - 1 1. 1 0.8 0.6 0.4 0.86 0.87 0.88 0.89 0.9 0.91 h (r.l.u.) Solid line --- fit to : 0. 0 0 1 3 4 5 6 7 8 9 ( ) ( 0) I H a is the only fitting parameter Best fit value - a =.4 with H I H (T) H 3.0H 1 a ln c = + Hc H c = 60T
Neutron scattering of La -xsrxcuo 4 at x=0.1 B. Lake, G. Aeppli, et al. H Hc Solid line - fit to : I( H) a ln H H = c
Presence of vortex lattice leads to supermodulation in the spin exciton spectrum Computation of spin susceptibility χ ( k, ω) in self-consistent large N theory of φ fluctuations in a vortex lattice α 0.8 0.7 r = 0.1, H/H c = 0.04 Intensity 0 ω 0.6 0.5 0.4 5.000 10.00 15.00 0.3 0.00 0. 5.00 6.00 0.1 0.0 0.0 0.5 1.0 1.5.0.5 3.0 3.5 k π / ( vortex lattice spacing)
Consequences of a finite London penetration depth (finite κ)
Further neighbor magnetic couplings La CuO4!" S 0 Conclusions Experiments Magnetic order!" S = 0 T=0 Concentration of mobile carriers δ Confined, paramagnetic Mott insulator has 1. Stable S=1 spin exciton. φ α. Broken translational symmetry:- bondcentered charge order. 3. Pairing of holes. 4. S=1/ moments near non-magnetic impurities Theory of magnetic ordering quantum transitions in antiferromagnets and superconductors leads to quantitative theories for Spin correlations in a magnetic field Effect of Zn/Li impurities on collective spin excitations